# Full text of "Analytical Mechanics"

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```MOTION OF A PARTICLE                       135
it.    Since the motion is along the x-axis the velocity has
dx
no components along the other axes, consequently v = -=--
ctz
Therefore equation (1) may be written in the form
™   ®~X   _         7   2
"I*    Jt*    ~~        ^    X>
or
k2 where o>2 = — -   Multiplying both sides of equation (2) by
THj
dx 2—dt and integrating
\dts or                                    v= v c2 — co2#2,
where c2 is the constant of integration.    Let v = VQ when x = 0, then c2 = vQ2.    Therefore
V = -%/?; 2__co2X2                                               (3)
In order to find the second integral of equation (2) rewrite equation (3) in the form
dx
dt
Separating the variables in the last equation and integrating we have
.   _, uX _             ,
or                                       x = — sin (at + c')
CO
= a sin (ut + cf), where c' is the constant of integration and a=~.   Let
CO
x = 0 when t = 0, then c' *= 0, and
x = a sin co£.                                (4)```